/**************************************************************************1thickness.m2William Stein3December 12, 199945There is a computation that might be easy for you to make, that might6shed light on whether or not the topological closure of the set of modular7Galois representations (in the universal deformation space) contains an8open set. Tell me whether or not this is feasible. Start, say, with the9Fourier expansion mod 3 of the newform of weight two of level 11 (call it10w) but I want to ignore the Euler factor at 3, i.e., I want to ignore the11Fourier coefficents a_n where n is divisible by 3. Now look for eigenforms12of arbitrary weight, of level 33, whose Fourier coefficients are in Z_3 and13are congruent to those of w modulo 3 (ignoring the Fourier coefficents a_n14where n is divisible by 3). Every time you find one, RECORD its Fourier15expansion modulo 9 (again ignoring the Fourier coefficents a_n where n is16divisible by 3). Count the number of these ( Fourier expansion modulo 917ignoring the Fourier coefficents a_n where n is divisible by 3) that you18get. Do you get 27 of them?1920***********************************************************************/2122232425/*26Step 1: Start, say, with the27Fourier expansion mod 3 of the newform of weight two of level 11 (call it28w) but I want to ignore the Euler factor at 3, i.e., I want to ignore the29Fourier coefficents a_n where n is divisible by 3.30*/3132prec := 37;33"Precision: ", prec;3435function To_pAdic(g,p,q)36return &+[pAdicRing(p)!(Integers()!Coefficient(g,n))*q^n37: n in [1..Degree(g)]];38end function;3940R<q> := PowerSeriesRing(pAdicRing(3));4142w := To_pAdic(qEigenform(ModularFactor("11k2A"),prec), 3, q);43"w = ", w;4445/*46Step 2: Now look for eigenforms47of arbitrary weight, of level 33, whose Fourier coefficients are in Z_3 and48are congruent to those of w modulo 3 (ignoring the Fourier coefficents a_n49where n is divisible by 3).50*/515253function ValPrimeTo(f,p)54if f eq 0 then55return 99999999999;56end if;57v, pos := Min([Valuation(Coefficient(f,i)) : i in [1..Degree(f)]58| (i mod p ne 0) and Valuation(Coefficient(f,i)) ne 0]);59return v;60end function;6162function IsCongruentToW(f)63return ValPrimeTo(f-w,3) ge 1;64end function;6566function pDeprive(f,p)67R<q> := Parent(f);68for i in [1..Ceiling(Degree(f)/3)] do69f -:= Coefficient(f,3*i)*q^(3*i);70end for;71return f;72end function;7374function Level33Search(k1, k2)75R9 := PowerSeriesRing(Integers(9));76RZ := PowerSeriesRing(Integers());77nearby_forms := [R9|];78for k in [w : w in [k1..k2] | IsEven(w)] do79"k = ",k;80v := ZpRationalNewforms(11,k,3,prec)81cat ZpRationalNewforms(33,k,3,prec);82"v = ", v;83for f in v do84if IsCongruentToW(f) then85g := pDeprive(R9!(RZ!f),3);86Append(~nearby_forms, g);87"g = ",g;88end if;89end for;90end for;91return nearby_forms;92end function;93